recent advances in elliptic complex geometryforstneric/datoteke/rafrot-2010.pdf · stein manifolds...

Recent advances in elliptic complex geometry

Franc Forstneric

University of Ljubljana

RAFROT, Rincon, March 22, 2010

Stein manifolds

A Stein manifold is a complex manifold S with many holomorphicfunctions S → C.Function-theoretically, S is similar to domains in C.

Stein manifolds


Examples: Domains in C.Open Riemann surfaces.Domains of holomorphy in C

n.Closed complex submanifolds of C

N .

Stein manifolds




N .

What is the right way to interpret “many functions”?

Stein manifolds




N .


Several equivalent definitions:

• S is holomorphically convex and holomorphic functions separatepoints and provide local charts.

Stein manifolds




N .




• S is a closed complex submanifold of CN , some N.

Stein manifolds




N .





• Hk(S , F ) = 0 for all coherent analytic sheaves F , k ≥ 1.

Stein manifolds




N .





• Hk(S , F ) = 0 for all coherent analytic sheaves F , k ≥ 1.

• S has a strictly plurisubharmonic exhaustion function.

What should the dual notion be?

Some important complex manifolds X admit many holomorphicmaps C → X , and C

k → X ; for example Cn, P

n, complex Liegroups and their homogeneous spaces.





Opposite properties: Kobayashi-Eisenman-Brody hyperbolicity (nononconstant maps C → X , no maps C

k → X of maximal rank,...)







Can we find a good way to interpret “many”?








Start with two classical theorems of 19th century complex analysis.









Weierstrass Theorem. On a discrete subset of a domain Ω in C,we can prescribe the values of a holomorphic function on Ω.









Weierstrass Theorem. On a discrete subset of a domain Ω in C,we can prescribe the values of a holomorphic function on Ω.

Runge Theorem. Let K ⊂ C be compact with no holes. Everyholomorphic function K → C can be approximated uniformly on Kby entire functions.

Cartan and Oka-Weil

Higher-dimensional generalisations:

Cartan and Oka-Weil


Cartan Extension Theorem. If T is a subvariety of a Steinmanifold S , then every holomorphic function T → C extends to aholomorphic function S → C.

Cartan and Oka-Weil



Oka-Weil Approximation Theorem. Let K be a holomorphicallyconvex compact subset of a Stein manifold S (i.e., for every pointp ∈ S\K there exists f ∈ O(S) such that |f (p)| > supK |f |). Thenevery holomorphic function K → C can be approximated uniformlyon K by holomorphic functions S → C.

Cartan and Oka-Weil




These are fundamental properties of Stein manifolds.

Cartan and Oka-Weil





We can also view them as properties of the target C.

Cartan and Oka-Weil





We can also view them as properties of the target C.

Formulate them as properties of an arbitrary target X , with sourceany Stein manifold (or Stein space).

Oka properties of a complex manifold

A property that a complex manifold X might or might not have:

Basic Oka Property (BOP): For every Stein inclusion T → Sand every compact O(S)-convex set K ⊂ S , a continuous mapf : S → X that is holomorphic on K ∪ T can be deformed to aholomorphic map F : S → X .The deformation (homotopy) can be kept fixed on T andholomorphic on K (approximating f on K ).




T

f // X

S

F

>>~~~~~~~~




T

f // X

S

F

>>~~~~~~~~

By Oka-Weil and Cartan, C, and hence Cn, satisfy BOP.

No hyperbolic (or volume hyperbolic) X has BOP.

Parametric Oka Property (POP)

Let Q ⊂ P be compacts in Rm. Consider maps f : P × S → X .



Q //

O(S , X ) //

C (S , X )

P //

44jjjjjjjjjjj

;;

O(T , X ) // C (T , X )

POP of X : Every lifting in the big square can be deformed throughsuch liftings to a lifting in the left-hand square.



Q //

O(S , X ) //

C (S , X )

P //

44jjjjjjjjjjj

;;

O(T , X ) // C (T , X )


Applying POP with parameter pairs ∅ ⊂ Sk and Sk ⊂ Bk+1 showsthat for any POP manifold X and for any Stein manifold S :

πk(O(S , X )) ∼= πk(C(S , X )), ∀k = 0, 1, 2, . . .



Q //

O(S , X ) //

C (S , X )

P //

44jjjjjjjjjjj

;;

O(T , X ) // C (T , X )


Applying POP with parameter pairs ∅ ⊂ Sk and Sk ⊂ Bk+1 showsthat for any POP manifold X and for any Stein manifold S :

πk(O(S , X )) ∼= πk(C(S , X )), ∀k = 0, 1, 2, . . .

The isomorphisms of homotopy groups are induced by the inclusionO(S , X ) → O(S , X ).

The Oka-Grauert Principle

Theorem (Grauert, 1957-58): Every complex homogeneousmanifold enjoys POP for all pairs of finite polyhedra Q ⊂ P.The analogous result holds for sections of holomorphic G -bundles(G a complex Lie group) over a Stein space.



Since every isomorphism between G -bundles is a section of anassociated G -bundle, we get

Corollary. The holomorphic and topological classifications of suchbundles over Stein spaces coincide.This holds in particular for complex vector bundles.



Since every isomorphism between G -bundles is a section of anassociated G -bundle, we get

Corollary. The holomorphic and topological classifications of suchbundles over Stein spaces coincide.This holds in particular for complex vector bundles.

Kiyoshi Oka proved this result for line bundles in 1939: A Cousin-IIproblem is solvable by holomorphic functions if it is solvable bycontinuous functions.

Elliptic manifolds

Gromov, 1989: A complex manifold X is elliptic if it admits adominating spray:A holomorphic map s : E → X defined on the total space of aholomorphic vector bundle E over X such that s(0x) = x ands|Ex → X is a submersion at 0x for all x ∈ X .

Elliptic manifolds


b

b

be s

Xx

Ex

Elliptic manifolds


b

b

be s

Xx

Ex

Subelliptic manifold (F. 2002): There exist finitely many sprayssj : Ej → X such that

∑

j

dsj(Ej ,x) = TxX , ∀x ∈ X .

Examples of (sub) elliptic manifolds

• A homogeneous X is elliptic: X × g →s X , (x , v) 7→ exp(v) · x .



• Assume that X admits C-complete holomorphic vector fieldsv1, . . . , vk that span TxX at every point. Let φ

jt denote the flow of

vj . Then the map s : E = X × Ck → X ,

s(x , t1, . . . , tk) = φ1t1 · · · φk

tk(x)

is a dominating spray on X .






s(x , t1, . . . , tk) = φ1t1 · · · φk

tk(x)


• A spray of this type exists on X = Cn\A where A is algebraic

subvariety with dimA ≤ n − 2.Use shear vector fields f (π(z))v (v ∈ C

n, π : Cn → C

n−1 linearprojection, π(v) = 0) that vanish on A: f = 0 on π(A) ⊂ C

n−1.






s(x , t1, . . . , tk) = φ1t1 · · · φk

tk(x)




n, π : Cn → C


n−1.

• Pn\A is subelliptic if A is a subvariety of codimension ≥ 2.






s(x , t1, . . . , tk) = φ1t1 · · · φk

tk(x)




n, π : Cn → C


n−1.

• Pn\A is subelliptic if A is a subvariety of codimension ≥ 2.

Problem: Lack of known functorial properties of (sub)ellipticity

Gromov’s Oka principle

Theorem (Gromov 1989). POP holds in the following cases:1. Maps S → X from a Stein manifold S to an elliptic manifold X .2. Sections of a holomorphic fiber bundle Z → S with elliptic fiberover a Stein manifold S .3. Sections of an elliptic submersion Z → S over a Stein S .



Elliptic submersion: A holo. submersion Z → S with fiberdominating sprays over small open sets in S .




Detailed proofs: Forstneric & Prezelj, 2000-2002.





F. 2002: Sections of subelliptic submersions Z → S satisfy POP.






F. 2010: Sections of stratified subelliptic submersions over Steinspaces satisfy POP. (This includes stratified fiber bundles withsubelliptic fibers.)






F. 2010: Sections of stratified subelliptic submersions over Steinspaces satisfy POP. (This includes stratified fiber bundles withsubelliptic fibers.)

S = S0 ⊃ S1 ⊃ · · · ⊃ Sm = ∅, Mk = Sk\Sk+1 smooth, therestriction of Z |Mk a subelliptic submersion.

Sections avoiding subvarieties

Example: Let E → S be a holo. vector bundle with fiber Ex∼= C

k ,and let Σ ⊂ E be a tame complex subvariety with fibers Σx ⊂ Ex

of codimension ≥ 2. Then E\Σ → S is an elliptic submersion.Hence sections S → E avoiding Σ satisfy POP.





Tameness: The closure of Σ in the associated bundle E → S withfibers Ex

∼= Pk does not contain the hyperplane at infinity

Ex\Ex∼= P

k−1 over any point x ∈ X .

Algebraic subvarieties are tame.







Ex\Ex∼= P



Applications:Removal of intersections of maps S → C

n, Pn with algebraicsubvarieties of codim. ≥ 2. (Special case: complete intersections.)







Ex\Ex∼= P





Existence of proper holo. embeddings Sn → CN , N =

[3n2

]+ 1,

when Sn is Stein and n > 1.







Ex\Ex∼= P






[3n2

]+ 1,


Existence of proper holo. immersions Sn → CN , N =

[3n+1

2

].







Ex\Ex∼= P






[3n2

]+ 1,


Existence of proper holo. immersions Sn → CN , N =

[3n+1

2

].

H-principle for holomorphic immersions Sn → CN .

Oka manifolds

Oka manifolds

Gromov (1989): Can one characterize BOP and POP by a Rungeapproximation property for maps C

n → X? BOP=⇒POP ?

Oka manifolds



Convex Approximation Property (CAP) (F. 2005):Every holomorphic map K → X from a compact (geometrically!)convex set K ⊂ C

n can be approximated uniformly on K byholomorphic maps C

n → X .

Oka manifolds





n → X .

Observe that CAP equals BOP in the model case S = Cn, K a

convex set in Cn, and T = ∅.

Oka manifolds





n → X .



Theorem (F. 2005, 2006, 2009) CAP ⇐⇒ POP for any Steinspace S as source, and for any compacts Q ⊂ P ⊂ R

m.

Oka manifolds





n → X .




m.

Corollary. All Oka type properties of a complex manifold areequivalent. Such manifolds are called Oka manifolds.

Oka manifolds





n → X .




m.

Corollary. All Oka type properties of a complex manifold areequivalent. Such manifolds are called Oka manifolds.

F. Larusson: What is an Oka manifold?Notices Amer. Math. Soc. 57 (2010), no. 1, 50–52.

Properties and examples

Theorem. If E → B is a holomorphic fiber bundle whose fiber isan Oka manifold, then B is Oka if and only if E is Oka.



Examples of Oka manifolds:

Cn, Pn, complex Lie groups and their homog. spaces





Cn\A, A tame analytic subvariety of codim. ≥ 2






Pn\A, A subvariety of codim. ≥ 2







Hirzebruch surfaces (P1 bundles over P1)








Hopf manifolds (quotients of Cn\0 by a cyclic group)









Algebraic manifolds that are locally Zariski affine (∼= Cn);










certain modifications of such (blowing up points, removingsubvarieties of codim. ≥ 2)










certain modifications of such (blowing up points, removingsubvarieties of codim. ≥ 2)C

n blown up at all points of a tame discrete sequence











n blown up at all points of a tame discrete sequencecomplex torus of dim> 1 with finitely many points removed, orblown up at finitely many points











n blown up at all points of a tame discrete sequencecomplex torus of dim> 1 with finitely many points removed, orblown up at finitely many points

Question: Is every Oka manifold also elliptic? (Gromov: EveryStein Oka manifold is elliptic.)

Methods to prove CAP =⇒ POP

A nonlinear Cousin-I problem



Let (A, B) be a Cartain pair in a Stein manifold S (compacts suchthat A ∪ B, A ∩ B have Stein neighborhood bases).




Given f : A → X , g : B → X holomorphic, with f ≈ g on A ∩ B,find a holomorphic map f : A ∪ B → X such that f |A ≈ f |A.





• Extend f , g to holomorphic maps

F : A × Bk → X , G : B × B

k → X ,

submersive in z ∈ Bk ⊂ C

k ; f = F (· , 0), g = G (· , 0).






F : A × Bk → X , G : B × B

k → X ,


k ; f = F (· , 0), g = G (· , 0).

• Find a holomorphic transition map γ(x , z) = (x , c(x , z)) over(A ∩ B) × rBk (r < 1), γ ≈ Id, such that F = G γ.






F : A × Bk → X , G : B × B

k → X ,


k ; f = F (· , 0), g = G (· , 0).

• Find a holomorphic transition map γ(x , z) = (x , c(x , z)) over(A ∩ B) × rBk (r < 1), γ ≈ Id, such that F = G γ.

• Splitγ = β α−1, α, β ≈ Id.

Then F α = G β : (A ∪ B) × rBk → X solves the problem.

Passing a critical point

Passing a critical point p0 of an exhaustion function ρ on S

ρ = −t0

Ωc

E

b

p0

ρ < c − t1ρ < c − t1

Figure: The set Ωc = τ < c, c > 0.

From manifolds to maps

Grothendieck: Properties of objects (manifolds, varieties) shouldgive rise to corresponding properties of maps (morphisms).



Oka properties of a holomorphic map π : E → B refer to liftings inthe following diagram, with Stein source S :

E

π

P × S

f //

F

<<xxxxxxxxx

B




E

π

P × S

f //

F

<<xxxxxxxxx

B

For a given S-holo. map f : P × S → B (with P a compact in Rm),

every continuous lifting F must be homotopic to an S-holo. lifting.




E

π

P × S

f //

F

<<xxxxxxxxx

B

For a given S-holo. map f : P × S → B (with P a compact in Rm),

every continuous lifting F must be homotopic to an S-holo. lifting.

Theorem (F. 2010) Let π : E → B a stratified holo. submersion.(a) BOP =⇒ POP, and these are local properties.(b) A stratified holo. fiber bundle with Oka fibers enjoys POP.(c) A stratified subelliptic submersion enjoys POP.

Recent application to Gromov-Vaserstein ProblemTheorem (Ivarsson and Kutzschebauch, 2009)Let S be a Stein manifold and f : S → SLm(C) a null-homotopicholomorphic mapping. There exist k ∈ N and holomorphicmappings G1, . . . ,Gk : S → C

m(m−1)/2 such that

f (x) =

(1 0

G1(x) 1

)(1 G2(x)0 1

). . .

(1 Gk(x)0 1

).



f (x) =

(1 0

G1(x) 1

)(1 G2(x)0 1

). . .

(1 Gk(x)0 1

).

Algebraic case: Consider algebraic maps Cn → SLm(C).



f (x) =

(1 0

G1(x) 1

)(1 G2(x)0 1

). . .

(1 Gk(x)0 1

).


Cohn (1966): The matrix(

1 − z1z2 z21

−z22 1 + z1z2

)∈ SL2(C[z1, z2])

does not decompose as a finite product of unipotent matrices.



f (x) =

(1 0

G1(x) 1

)(1 G2(x)0 1

). . .

(1 Gk(x)0 1

).



1 − z1z2 z21

−z22 1 + z1z2

)∈ SL2(C[z1, z2])


Suslin (1977): For m ≥ 3 (and any n) any matrix in SLm(C[n])decomposes as a finite product of unipotent matrices.



f (x) =

(1 0

G1(x) 1

)(1 G2(x)0 1

). . .

(1 Gk(x)0 1

).



1 − z1z2 z21

−z22 1 + z1z2

)∈ SL2(C[z1, z2])


Suslin (1977): For m ≥ 3 (and any n) any matrix in SLm(C[n])decomposes as a finite product of unipotent matrices.

Vaserstein (1988): Factorization of continuous maps.

Idea of proofDefine Ψk : (Cm(m−1)/2)k → SLm(C) by

Ψk(g1, . . . , gk) =

(1 0g1 1

)(1 g2

0 1

). . .

(1 gk

0 1

).


Ψk(g1, . . . , gk) =

(1 0g1 1

)(1 g2

0 1

). . .

(1 gk

0 1

).

We want to find a holomorphic mapG = (g1, . . . , gk) : S → (Cm(m−1)/2)k such that the followingdiagram commutes:

(Cm(m−1)/2)k

Ψk

S

f//

G99tttttttttttSLm(C)


Ψk(g1, . . . , gk) =

(1 0g1 1

)(1 g2

0 1

). . .

(1 gk

0 1

).


(Cm(m−1)/2)k

Ψk

S

f//


Vaserstein’s result gives a continuous lifting of f .


Ψk(g1, . . . , gk) =

(1 0g1 1

)(1 g2

0 1

). . .

(1 gk

0 1

).


(Cm(m−1)/2)k

Ψk

S

f//


Vaserstein’s result gives a continuous lifting of f .We deform this continuous lifting to a holomorphic lifting byapplying the Oka principle to certain auxiliary submersions (rowprojections SLm(C) → C

n) that are stratified elliptic.

A few open problems• Find a geometric characterisation of Oka manifolds. Clarify therelationship with Gromov’s ellipticity.


Larusson: Every quasi-projective algebraic X admits an affinebundle E → X with Stein total space. If X is Oka then E is elliptic.



• Which complex surfaces of non-general type are Oka?




• Is Cn\(closed ball) Oka? It contains biholomorphic images of C

n.





n.

• Let g : D = z ∈ C : |z | < 1 → C be a continuous map.Assume that π : Eg = D × C\Γg → D is Oka. Then there is aholomorphic map F : D × C

∗ → Eg such that the dgm. commutes:

D × C∗

proj##GG

GGGG

GGGG

F // Eg

π

D





n.



D × C∗

proj##GG

GGGG

GGGG

F // Eg

π

D

g(z) is the missing value in the range of the map F (z , · ) : C∗ → C.





n.



D × C∗

proj##GG

GGGG

GGGG

F // Eg

π

D

g(z) is the missing value in the range of the map F (z , · ) : C∗ → C.

Question: Must g be holomorphic?

Link with homotopy theory

Larusson 2004: POP is a homotopy-theoretic property.

The category of complex manifolds can be embedded into a modelcategory such that:• a holomorphic map is acyclic iff it is topologically acyclic.• a Stein inclusion is a cofibration.• a holomorphic map is a fibration iff it is a topological fibrationand satisfies POP.

Link with homotopy theory

Larusson 2004: POP is a homotopy-theoretic property.

The category of complex manifolds can be embedded into a modelcategory such that:• a holomorphic map is acyclic iff it is topologically acyclic.• a Stein inclusion is a cofibration.• a holomorphic map is a fibration iff it is a topological fibrationand satisfies POP.

Theorem (Larusson 2004–5). In this model structure, a complexmanifold is:• cofibrant iff it is Stein.• fibrant iff it has POP.

recent advances in elliptic complex geometryforstneric/datoteke/rafrot-2010.pdf · stein manifolds...

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